Free Banach lattices under convexity conditions

Abstract

We prove the existence of free objects in certain subcategories of Banach lattices, including p-convex Banach lattices, Banach lattices with upper p-estimates, and AM-spaces. From this we immediately deduce that projectively universal objects exist in each of these subcategories, extending results of Leung, Li, Oikhberg and Tursi (Israel J. Math. 2019). In the p-convex and AM-space cases, we are able to explicitly identify the norms of the free Banach lattices, and we conclude by investigating the structure of these norms in connection with nonlinear p-summing maps.

Geometry of unit balls of free Banach lattices, and its applications

“…∈ E so that i e i p Proof. The equivalence (1) ⇔ (4) has appeared in [12]; we prove all implications for the reader's convenience.…”

“…In particular, they have been essential in resolving a question of J. Diestel: [4] provides an example of a free Banach lattice which is weakly compactly generated as a Banach lattice, but not in the Banach space sense. We refer the reader to [12] and [19] for more information about free lattices.…”